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  • Please,please help me - Complex numbers and quadratic equations - JEE Main-3

Let z is a complex number such that \left |z \right |=1  then  \frac{z}{z^{2}-1}  is  

  • Option 1)

    Purely real 

  • Option 2)

    Purely imaginary

  • Option 3)

    complex with non-zero real & imaginary part

  • Option 4)

    Can't be determined

 

Answers (1)

best_answer

\left | Z \right |=1\; \Rightarrow \; Z\bar{Z}=1 \; \left ( on\; savaring \right )

\therefore \frac{Z}{Z^{2}-1}=\frac{Z}{Z^{2}-Z\bar{Z}}=\frac{1}{Z-\bar{Z}}

Let\; Z=x+iy\; \; then\; \; \cdot \bar{Z}=x-iy

\therefore \frac{1}{Z-\bar{Z}}=\frac{1}{2iy}=\frac{-i}{2y}

 

Property of conjugate of complex number -

z\bar{z}=\left |z \right |^{2}

- wherein

  z=x+iy, \bar{z}=conjugate \: of\: z   

 \left |z \right |=\sqrt{x^{2}+y^{2}}

 

 


Option 1)

Purely real 

This is incorrect

Option 2)

Purely imaginary

This is correct

Option 3)

complex with non-zero real & imaginary part

This is incorrect

Option 4)

Can't be determined

This is incorrect

Posted by

Himanshu

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